The fact that this classification applies only to compact manifolds should not worry us too much. We will see in section REF  that the curvature radius of any of these geometries is larger than the diameter of the observable universe. Therefore any notions of closedness or periodicity on spatial section is largely irrelevant to our local patch of the observable universe. Due to the fact inflation exponentially enlarges spatial sections of the Universe, it is highly likely that
any complex geometric structure of the very early Universe
is hidden behind the observable patch of the Universe, if we assume that cosmic inflation occurred.
Therefore it is reasonable to assume that the observable patch
of the Universe corresponds to one of the Thurston geometries,
and that any subtleties arising due superimposed topology
is not observable. Nevertheless, from the phenomenological point of
view, it is worth investigating the signatures of topology of the
Universe and confront them with the data. In fact, there is a large body of work mentioned above and doing precisely that, for reviews see
Refs. [1]}, [2]}, [3]}, [4]}, [5]}.
All of these works assume that the geometry of the spatial section
is flat, i.e. \(\Sigma _3 = {R}^3\) .
The question of how topology may affect
the analyses performed in this work we will not focus on.
